[This is akin to a living review, which will hopefully improve from time to time. Last edited 2017-11-26.]
This post will collect some models of decoherence and branching. We don’t have a rigorous definition of branches yet but I crudely define models of branching to be models of decoherenceI take decoherence to mean a model with dynamics taking the form for some tensor decomposition , where is an (approximately) stable orthonormal basis independent of initial state, and where for times and , where is the initial state of and is some characteristic time scale.a which additionally feature some combination of amplification, irreversibility, redundant records, and/or outcomes with an intuitive macroscopic interpretation. I have the following desiderata for models, which tend to be in tension with computational tractability:
symmetric (e.g., translationally)
no ad-hoc system-environment distinction
Ehrenfest evolution along classical phase-space trajectories (at least on Lyapunov timescales)
Regarding that last one: we would like to recover “classical behavior” in the sense of classical Hamiltonian flow, which (presumably) means continuous degrees of freedom.In principle you could have discrete degrees of freedom that limit, as , to some sort of discrete classical systems, but most people find this unsatisfying.b Branching only becomes unambiguous in some large-N limit, so it seems satisfying models are necessarily messy and difficult to numerically simulate.… [continue reading]
A senior colleague asked me for thoughts on this paper describing a single-preferred-branch flavor of quantum mechanics, and I thought I’d copy them here. Tl;dr: I did not find an important new idea in it, but this paper nicely illustrates the appeal of Finkelstein’s partial-trace decoherence and the ambiguity inherent in connecting a many-worlds wavefunction to our direct observations.
We start by assuming that a precise wavefunction branch structure has been specified. The idea, basically, is to randomly draw a branch at late times according to the Born probability, then to evolve it backwards in time to the beginning of the universe and take that as your initial condition. The main motivating observation is that, if we assume that all branch splittings are defined by a projective decomposition of some subsystem (‘the system’) which is recorded faithfully elsewhere (‘the environment’), then the lone preferred branch — time-evolving by itself — is an eigenstate of each of the projectors defining the splits. In a sense, Weingarten lays claim to ordered consistency [arxiv:gr-qc/9607073] by assuming partial-trace decoherenceNote on terminology: What Finkelstein called “partial-trace decoherence” is really a specialized form of consistency (i.e., a mathematical criterion for sets of consistent histories) that captures some, but not all, of the properties of the physical and dynamical process of decoherence. That’s why I’ve called it “partial-trace consistency” here and here.a [arXiv:gr-qc/9301004]. In this way, the macrostate states stay the same as normal quantum mechanics but the microstates secretly conspire to confine the universe to a single branch.
I put proposals like this in the same category as Bohmian mechanics. They take as assumptions the initial state and unitary evolution of the universe, along with the conventional decoherence/amplification story that argues for (but never fully specifies from first principles) a fuzzy, time-dependent decomposition of the wavefunction into branches.… [continue reading]
Here is an underemphasized way to frame the relationship between trajectories and symmetries (in the sense of Noether’s theorem)You can find this presentation in “A short review on Noether’s theorems, gauge symmetries and boundary terms” by Máximo Bañados and Ignacio A. Reyes (H/t Godfrey Miller).a . Consider the space of all possible trajectories for a system, a real-valued Lagrangian functional on that space, the “directions” at each point, and the corresponding functional gradient in each direction. Classical solutions are exactly those trajectories such that the Lagrangian is stationary for perturbations in any direction , and continuous symmetries are exactly those directions such that the Lagrangian is stationary for any trajectory . That is,
There are many subtleties obscured in this cartoon presentation, like the fact that a symmetry , being a tangent direction on the manifold of trajectories, can vary with the tangent point it is attached to (as for rotational symmetries). If you’ve never spent a long afternoon with a good book on the calculus of variations, I recommend it.
[Other parts in this series: .]
You’re taking a vacation to Granada to enjoy a Spanish ski resort in the Sierra Nevada mountains. But as your plane is coming in for a landing, you look out the window and realize the airport is on a small tropical island. Confused, you ask the flight attendant what’s wrong. “Oh”, she says, looking at your ticket, “you’re trying to get to Granada, but you’re on the plane to Grenada in the Caribbean Sea.” A wave of distress comes over your face, but she reassures you: “Don’t worry, Granada isn’t that far from here. The Hamming distance is only 1!”.
After you’ve recovered from that side-splitting humor, let’s dissect the frog. What’s the basis of the joke? The flight attendant is conflating two different metrics: the geographic distance and the Hamming distance. The distances are completely distinct, as two named locations can be very nearby in one and very far apart in the other.
Now let’s hear another joke from renowned physicist Chris Jarzynski:
The linear Schrödinger equation, however, does not give rise to the sort of nonlinear, chaotic dynamics responsible for ergodicity and mixing in classical many-body systems. This suggests that new concepts are needed to understand thermalization in isolated quantum systems. – C. Jarzynski, “Diverse phenomena, common themes” [PDF]
Ha! Get it? This joke is so good it’s been told by S. Wimberger“Since quantum mechanics is the more fundamental theory we can ask ourselves if there is chaotic motion in quantum systems as well. A key ingredient of the chaotic phenomenology is the sensitive dependence of the time evolution upon the initial conditions.… [continue reading]
Here is the first result out of the project Verifying Deep Mathematical Properties of AI SystemsTechnical abstract available here. Note that David Dill has taken over as PI from Alex Aiken.a funded through the Future of Life Institute.
You can find discussion on HackerNews. The lead author was kind enough to answers some questions about this work.
Q: Is the correctness specification usually a fairly singular statement? Or will it often be of the form “The program satisfied properties A, B, C, D, and E”? (And then maybe you add “F” later.)
Daniel Selsam: There are a few related issues: how singular is a specification, how much of the functionality of the system is certified (coverage), and how close the specification comes to proving that the system actually does what you want (validation).
Singular vs plural. If you want to certify a red-black tree, then you will probably want to prove many different properties about how the core methods (e.g. finding, inserting, deleting) interact, and so the specification will be rather plural. But a different system may use the certified red-black tree to do one very specific thing and may have a singular specification. Thus how singular or plural a specification is depends heavily on where we draw the boundaries between systems and is somewhat arbitrary. One way or another, the internals of any proof of correctness will need to make use of many different lemmas; sometimes you can tie them all up in a bow for a particular project and sometimes you cannot.
Coverage. In Certigrad, we prove that the sampled gradients are unbiased estimates of the true gradients, which arguably constitutes total functional correctness for the stochastic backpropagation algorithm.… [continue reading]
As has been discussed here before, the Reeh–Schlieder theorem is an initially confusing property of the vacuum in quantum field theory. It is difficult to find an illuminating discussion of it in the literature, whether in the context of algebraic QFT (from which it originated) or the more modern QFT grounded in RG and effective theories. I expect this to change once more field theorists get trained in quantum information.
The Reeh–Schlieder theorem states that the vacuum is cyclic with respect to the algebra of observables localized in some subset of Minkowski space. (For a single field , the algebra is defined to be generated by all finite smearings for with support in .) Here, “cyclic” means that the subspace is dense in , i.e., any state can be arbitrarily well approximated by a state of the form with . This is initially surprising because could be a state with particle excitations localized (essentially) to a region far from and that looks (essentially) like the vacuum everywhere else. The resolution derives from the fact the vacuum is highly entangled, such that the every region is entangled with every other region by an exponentially small amount.
One mistake that’s easy to make is to be fooled into thinking that this property can only be found in systems, like a field theory, with an infinite number of degrees of freedom. So let me exhibitMost likely a state with this property already exists in the quantum info literature, but I’ve got a habit of re-inventing the wheel. For my last paper, I spent the better part of a month rediscovering the Shor code…a a quantum state with the Reeh–Schlieder property that lives in the tensor product of a finite number of separable Hilbert spaces:
As emphasized above, a separable Hilbert space is one that has a countable orthonormal basis, and is therefore isomorphic to , the space of square-normalizable functions.… [continue reading]